Optical scheme for generating hyperentanglement having photonic qubit and time-bin via quantum dot and cross-Kerr nonlinearity

We design an optical scheme to generate hyperentanglement correlated with degrees of freedom (DOFs) via quantum dots (QDs), weak cross-Kerr nonlinearities (XKNLs), and linearly optical apparatuses (including time-bin encoders). For generating hyperentanglement having its own correlations for two DOFs (polarization and time-bin) on two photons, we employ the effects of optical nonlinearities using a QD (photon-electron), a parity gate (XKNLs), and time-bin encodings (linear optics). In our scheme, the first nonlinear multi-qubit gate utilizes the interactions between photons and an electron of QD confined in a single-sided cavity, and the parity gate (second gate) uses weak XKNLs, quantum bus, and photon-number-resolving measurement to entangle the polarizations of two photons. Finally, for efficiency in generating hyperentanglement and for the experimental implementation of this scheme, we discuss how the QD-cavity system can be performed reliably, and also discuss analysis of the immunity of the parity gate (XKNLs) against the decoherence effect.

For the feasibility and efficiency of quantum information processing, schemes that could realize quantum information processing should be designed using physical resources and experimental implementation. From this point of view, quantum optics assisted by optical nonlinearities plays a significant role in experimentally realizing quantum information processing.
In Fig. 1(b), when the left circularly polarized photon | 〉 L (right circularly polarized photon | 〉 R ) is injected into the QD-cavity system, the polarized photon can create the spin state ↑↓ ↓↑ ( ) coupled to X − in the spin state ↑ ↓ ( ) of the excess electron in QD according to the Pauli exclusion principle. By these spin-dependent optical transitions, the hot cavity (| 〉 ↓ R or | 〉 ↑ L : the QD is coupled to the cavity) and the cold cavity (| 〉 ↑ R or | 〉 ↓ L : the QD is uncoupled from the cavity) can induce different reflectances [|r h (ω)|, |r 0 (ω)|] and phases [ϕ rh (ω) = arg(r h (ω)), ϕ r0 (ω) = arg(r 0 (ω))] of the reflected photon, as follows:  where r h (ω) and r 0 (ω) are the reflection coefficients, ω c and ω are the frequencies of cavity mode and external field, and κ and g are the cavity decay rate of the cavity mode and the coupling strength (X − ↔ cavity mode). In the weak excitation approximation 77  depending on the interaction between a polarized photon and the spin state of an electron inside a single-sided cavity, is given by Here, if the QD-cavity system having the small side-leakage rate, κ s (κ κ  s ), the strongly coupling strength κ γ  g ( , ) with small γ (about several μeV) for ω ω = − c X , as shown in 33,78-80 , we can acquire |r 0 (ω)| = |r h (ω)| ≈ 1, ϕ rh (ω) = 0, and ϕ r0 (ω) = ±π/2 through adjustment of the frequencies between the external field and cavity mode (ω ω κ − =  /2 c ), and by omitting the leaky modes Ŝ in and vacuum noise N 20,33-40 . Finally, when we take the experimental parameters g/κ = 2.4 and κ s = 0 (negligible) with ω − ω c = κ/2 and γ/κ = 0.1, the reflection operator ω R( ) in Eq. 2 can be expressed as Subsequently, we will utilize this interaction of the QD-cavity system as a nonlinear optical device for the generation of hyperentanglement in our scheme.
The polarization entangler (parity gate). The XKNL's Hamiltonian is given as H Kerr = ħχN 1 N 2 , where N i is photon-number operator and χ is the magnitude of nonlinearity in the Kerr medium. If we consider |n〉 1 (photon state: n means photon-number) and |α〉 2 (coherent state or probe beam), the state of photon-probe system is transformed to α α α = = after the interactions in the Kerr medium, where θ = χt and t are the conditional phase-shift and the interaction time. Figure 2 shows a parity gate (polarization entangler), which can be operated using XKNLs, quantum bus beams, and photon-number-resolving measurement, to create entanglement between the polarizations of two photons. This parity gate 10,60,61,64,68,74 is composed of four polarizing beam splitters (PBSs), four conditional phase-shifts (positive phases) in Kerr media, two linear phase-shift (minus phase), and two beam splitters (BSs) in quantum bus beams, as described in Fig. 2.
Here, let us assume the input state is   When the measurement outcome is | 〉 0 b (dark detection), the output state will be | 〉 via feed-forward [shifting of relative phase by PS (Φ n )] in accordance of measurement outcome n in Fig. 2. The error probability (P err ), which is the probability of detection as | 〉 0 b (dark detection) in | 〉 n b on path b, of this parity gate can be calculated by , where sin 2 θ ≈ θ 2 for the strong magnitude of coherent state (probe beam: α  1) and θ  1. This means that the error probability, P err , can approach zero when increasing the magnitude of the probe beam or the conditional phase-shift (θ) of XKNL. However, the magnitude of XKNLs is tiny (very weak: θ ≈ 10 −18 ) 81 , although the magnitude of the conditional phase-shift could be enhanced by electromagnetically induced transparency (EIT), θ ≈ 10 −2 82,83 . Furthermore, as ref. 84 , it is experimentally difficult to implement the minus conditional phase-shift in the XKNL. Thus, we will utilize this polarization entangler (parity gate) employing quantum bus beams and photon-number-resolving measurement, which requires no negative XKNL (−θ) as a nonlinear optical device for the generation of hyperentanglement in our scheme.

Scheme of generating hyperentanglement in two photons using a QD-cavity system, parity gate, and time-bin encoders
We represent the generation of hyperentanglement having correlations for two DOFs (polarization and time-bin) via nonlinear optical devices (the QD-cavity system and parity gate) and linear optical apparatuses (including time-bin encoders), as shown in Fig. 3.
For a detailed description of this procedure, we assume the initial state of two photons as If the polarization of photon A is a vertically polarized state, | 〉 V , then we perform the spin-flipper (SF) by feed-forward before the polarization entangler, as described in Fig. 3. After (1) time-bin encoder, the state ϕ | 〉 1 AB of two photons is given by where the path length of photon B is longer than the path of photon A by DL (photon A: time interval s in the short length, photon B: time interval l in the long length) before CPBSs. Then, we can adjust the paths regarding two circular polarizations |R〉 and |L〉 of two photons by CPBSs, such as |L〉 obtaining the time interval l in the long length and |R〉 obtaining the time interval s over the short length. Finally, photon A (B) can obtain the time interval s (l) in the optical length, because the path length of photon A is shorter than the path of photon B before Switch 1 (S1). Subsequently, we prepare a spin state, , of electron 1 in QD. After the photons A and B pass through S1 according to the time-table of switches in Fig. 3, they interact with the QD-cavity system, in sequence. The output state ϕ | 〉 2 1AB of electron 1 and two photons, after the interactions between photons and an electron of QD as described in Eq. 3, is transformed as below where we consider that the total time interval of the path length and the interaction time of the QD-cavity is q, and also that |−〉 ≡ ↑ − ↓ ( ) / 2 . In (2) the time-bin encoder, after the photons A and B pass through S2 due to the time-table of switches in Fig. 3, photon A (B) can obtain the time interval l (s) in the optical length because the path length of photon A is longer than the path of photon B before PCs. Then we utilize PCs, which affect a bit-flip operation on the polarization at a specific time 10,16,20,23,75 , to flip the polarizations of the photons. Here the action of the PCs flips the polarizations of the photon A at time-bin qssll(=lqssl), and the photon B at time-bin qslll(=sqlll). After PCs, the photon A (B) can acquire the time interval l (s) in the optical length because the path length of photon B is shorter than the path of photon A by DL. Thus, after passing the (2) time-bin encoder, the output state is given by 2 1AB (2) time bin encoder, SF (photon B): feed forward 2 AB1 Because the initial state of photon A is a vertically polarized state, |V〉 A , we perform SF to photon B by feed-forward | 〉 → | 〉 R L ( ) B B , as described in Fig. 3 before the polarization entangler. In the polarization entangler (parity gate) using XKNLs, as described in Sec. 2, the output state ϕ | 〉 f 1AB , according to Eq. 4, is transformed, as follows: where we define the four types of polarization state (two photons), and the time-bin state (time interval), as follows: where {|H〉, |V〉} is the horizontal, vertical polarization on the photon, and {|s′〉, |l′〉}is short interval, long interval due to the path length of the photon. Subsequently, if the measurement outcomes of the QD-cavity system (electron 1), and the quantum bus beam on path b by the photon-number-resolving measurement are , then the final hyperentangled state having its own correlations for two DOFs (polarization and time-bin) on two photons will be Φ Ψ ⊗ in Eq. 8 is transformed to Φ − P AB via feed-forward [shifting of relative phase by PS (Φ n )] according to result n. Table 1 shows that the possible hyperentangled states (having two DOFs) of two photons can be generated in accordance with the preparation of the initial states (product state), and the measurement outcomes of electron So far, we designed a scheme to generate hyperentanglement having its own correlations for two DOFs (polarization and time-bin) utilizing nonlinear optical devices (the QD-cavity system and the parity gate using XKNLs), and the linear optical apparatuses (time-bin encoders). In our schemes, the important parts (nonlinear optical devices) are the QD-cavity system and polarization entangler (parity gate) utilizing XKNLs, quantum bus beams, and the photon-number-resolving measurement. Therefore, we will analyze the performance and efficiency of the nonlinear optical devices for the experimental implementation in practice.

Analysis of nonlinear parts: QD-cavity system and parity gate using XKNLs
The QD-cavity system. The QD-cavity system interactions, which can induce difference in the reflectances and phases of the reflected photon according to the hot cavity (coupled) and cold cavity (uncoupled) conditions in Eq. 1, are significantly utilized for the reliable performance of our scheme. Thus, we should analyze the actual efficiency and experimental performance of these interactions of the QD-cavity system. For the reflection coefficient r(ω) with the noise N(ω) and leakage S(ω) coefficients, the Heisenberg equations of motion for a cavity field operator â ( ), a dipole operator σ − ( ) of X − , and the input-output relations 77 , are given by where Ŝ in is an input field operator from leaky modes due to sideband leakage and absorption, and N is the vacuum noise operator for σ − . In the weak excitation approximation 77 and the ground state in QD (i.e., σ , we can calculate the reflection coefficient R(ω) with the noise N(ω) and leakage S(ω) coefficients, as follows: Considering the hot (g ≠ 0) and cold (g = 0) cavities, the reflection coefficients [r h (ω) and r 0 (ω)] are represented in Eq. 1. In addition, the noise [n h (ω) and n 0 (ω)] rates; and leakage [s h (ω) and s 0 (ω)] coefficients are given by  The initial state of two photons (product state)

Result of electron 1
Result of photon-numberresolving measurement
The polarization entangler (parity gate). In the parity gate using XKNLs, the decoherence effect gives rise to photon loss of quantum bus beams and dephasing of coherent parameters of photon-probe systems in optical fibers [69][70][71][72][73][74]85,86 . Thus, the probability of success (P suc = 1 − P err ) and the fidelity F XKNL of the output state between the ideal case and the practical case will decrease due to photon loss and dephasing (evolving quantum state to mixed state) in practice. For analysis of the influence of the decoherence effect, we introduce a master equation 87 to describe the parity gate.
where λ and t(= θ/χ) are the energy decay rate and the interaction time in the solution ρ ρ = +t J L t ( ) exp[( )] (0). Due to this solution, we can calculate the interaction of XKNL ∼ X t with the decoherence effect (photon loss and dephasing),   ( 1) where we consider the divided interaction time Δt(=t/N) with N = 10 3 and θ = χt = χNΔt = NΔθ for a good approximation of our analysis Moreover, Λ t = e −λt/2 is the photon decay rate (photon loss) after the probe beam emerges from the Kerr medium. When the parity gate is implemented in practice for generating the controlled phase shift (XKNL) θ = π, the requirement for the length of the optical fiber is about 3000 km, according to χ/λ = 0.0125 (0.364 dB/km) the signal loss of commercial fibers 85,86 , and χ/λ = 0.0303 (0.15 dB/km) of pure silica core fibers 86 . Therefore, we should consider the decoherence effect (photon loss and dephasing) in the experimentally realized parity gate via our analysis. By the modeling in Eq. 18 (from the master equation) and the practical optical fiber, the output state ϕ | 〉 RR in Eq. 4 of the parity gate, as described in Section 2 will evolve to a mixed state ρ RR , as follows: where the row and column of ρ RR are α The coherent parameters (C, M, L, O, and K) of the off-diagonal terms from Eq. 18 are given by 74  ( 1) ( ) Figure 6 represents the absolute values of coherent parameters (the off-diagonal terms in ρ RR , Eq. 19) with regard to the difference in the amplitudes of the probe beam, α with the fixed αθ = αχt = 2.5 for P err < 10 −3 in the optical fiber having a signal loss χ/λ = 0.0303 (0.15 dB/km) 70,71,74,86 .
In Fig. 6, we can conclude that the amplitude of the coherent state (probe beam α) should be increased to constrain the output state ρ RR to the pure state (coherent parameters approaching '1') for experimentally reliable performance of the parity gate. Thus, the reliable performance of the parity gate can be acquired by using a strong coherent state (probe beam) when the fixed αθ = 2.5 for P err < 10 −3 in the optical fiber, χ/λ = 0.0303 (0.15 dB/ km), according to our analysis via the master equation (Eq. 17).
Due to the above result, we calculate the fidelity F XKNL between the output state ϕ | 〉 RR , Eq. 4, in the ideal case and the output state ρ RR , Eq. 19, in the practical case. When we take the parameters N = 10 3 , αθ = αχt ≈ 2.5 for P err < 10 −3 in the optical fiber, χ/λ = 0.0303 (0.15 dB/km); the fidelity F XKNL is given by XKNL  2  2  2  2  2  2  2 where C, M, L, O, and K are coherent parameters (off-diagonal terms in ρ RR , Eq. 19) in Eq. 20. Figure 7 shows that we can obtain high fidelity for the output state (F XKNL → 1) by using a strong coherent state (probe beam) in the optical fiber, χ/λ = 0.0303 (0.15 dB/km) 86 when the fixed αθ = 2.5 for P err < 10 −3 . Furthermore, if the strong coherent state is utilized for efficient and reliable performance (high fidelity and the robustness from photon loss and dephasing induced by the decoherence effect) of the parity gate, this should decrease the magnitude of the conditional phase-shift by XKNL, as described in Fig. 7 (the left table). Therefore, we can improve the experimental feasibility of implementation of the parity gate because the natural XKNLs are extremely weak 81 . Consequently, this gate can be operated with reliable performance and the immunity from the decoherence effect for the generation of hyperentanglement in our scheme because this analysis of the parity gate using XKNLs, quantum bus beams, and the photon-number-resolving measurement.

Conclusions
We herein propose an optical scheme to generate hyperentanglement having its own correlations for two DOFs (polarization and time-bin) on two photons using a QD-cavity system, a parity gate (XKNLs), and linear optical apparatuses (including time-bin encoders). For the reliable performance of this scheme, the most important components are two nonlinear optical devices, such as the QD-cavity system (QD in a single-sided cavity) and the polarization entangler gate (parity gate via XKNLs).
From the results of our analysis (the QD-cavity system) in Sec. 4, when the coupling strength, g/κ, is strong κ γ  g ( ( , )), and κ s /κ is the small side leakage rate κ κ  ( ) s with ω − ω c = κ/2 (single-sided), we can acquire high fidelity F QD of the output state (photon-electron) with a negligible amount of leaky modes Ŝ in and vacuum noise N . For this result, many researches have been studied, as follows: For the experimental requirements (strong coupling strength and small side leakage) in practice, Reithmaier et al. 33 obtained the coupling strength κ κ + ≈ . g/( ) 0 5 s in a micropillar cavity at d = 1.5 μm for the quality factor Q = 8800. When Q = 40000, increasing the coupling strength as g/(κ + κ s ) ≈ 2.4 could be experimentally obtained as in 88 . For strong coupling, Bayer et al. 89 demonstrated that micropillars with d = 1.5 μm and γ/κ ≈ 1 μeV (the decay rate of X − ) could be acquired from In 0.6 Ga 0.4 As/GaAs (QDs) with the temperature T ≈ 2 K 89 . The side leakage rate κ s can be reduced by optimizing the etching process (or improving the sample growth) with g/(κ + κ s ) ≈ 2.4, when g ≈ 80 μeV and Q = 40000 (including the side leakage rate κ s ) have been realized with In 0.6 Ga 0.4 As 79 . Moreover, the small side leakage rate can be obtained by improving the quality factor to Q = 215000 (κ ≈ 6.2 μeV) 90 .  The left table is a list of the values of F XKNL for α = 500, 10 3 , and 10 5 in the optical fiber having χ/λ = 0.0303 (0.15 dB/km). The right plot represents F XKNL of the output state with respect to differences in the amplitude of the coherent state (α) and the signal loss χ/λ in the optical fiber. Here we take the parameter αθ = 2.5 for the error probability P err < 10 −3 .
Scientific REPORTS | (2018) 8:2566 | DOI:10.1038/s41598-018-19970-2 In the case of the parity gate using XKNLs, for immunity (F XKNL → 1) against the decoherence effect, we utilize quantum bus beams and photon-number-resolving measurement with the strong coherent state according to our analysis in Section 4. The multi-qubit gates using homodyne measurements 60,64,68 cannot prevent evolution of the result (pure) state to a mixed state induced by the decoherence effect [69][70][71][72][73][74] , and also require a minus conditional phase-shift, −θ (which is not easy to implement due to 84 ). However, our parity gate (polarization entangler) via XKNLs, quantum bus beams, and the photon-number-resolving measurement has the following advantages: First, the acquisition of robustness (preventing the dephasing of coherent parameters in Fig. 6) against the decoherence effect is possible to utilize the strong coherent state (probe beam) via our analysis using the master equation, as described in Section 4. Second, there is no requirement for the minus conditional phase-shift, −θ, which is generally known as the impossibility of changing the sign of the conditional phase-shift 84 . Third, the feasibility and experimental realization (the natural XKNLs are extremely weak 81 ) are enhanced. This is because we can much reduce the magnitudes of the conditional phase-shifts, as listed in Fig. 7, if we employ the strong coherent state for the suppression of decoherence effect (preventing the photon loss and dephasing).
Consequently, we designed our scheme to generate hyperentanglement on two DOFs (polarization and time-bin) of two photons via the QD-cavity system, and the parity gate (polarization entangler) using XKNL, quantum bus beams, photon-number-resolving measurement, and linear optical apparatuses (time-bin encoders). Furthermore, we demonstrated by analysis the efficiency (with performance) and experimental feasibility of the nonlinear parts [QD-cavity system and parity gate (XKNLs)], which are critical components in our scheme to reliably generate hyperentanglement.